Boundaries of flow change in an array of structures with an inclined slope

. The purpose of the research is to substantiate an effective technology for the use of reinforced soil in hydraulic engineering facilities. The main task is to develop calculation methods and compare them with the results of model studies. The article uses methods of physical modeling of structures made of reinforced soil. It develops the theoretical foundations of modeling by introducing the obtained data on full-scale structures. The values of the dimensionless coefficients and the corresponding hydraulic gradients for various times are given in the tabular form, which is convenient for calculating operational parameters. To clarify the change in the boundary of the filtration flow over time, experiments were carried out in a slotted tray. The article graphically depicts the results of theoretical studies and experiments; the discrepancy between the theoretical and experimental curves for changing the boundary of the filtration flow over time is no more than 4%.


Introduction
In an array of structures with an inclined slope, including channels and other objects of hydropower systems and machine water lifting systems, a filtration flow is formed with the possibility of slope deformation.Calculations of water movement in these systems are associated with satisfying the demands of various sectors of the national economy, which do not allow the possibility of disrupting the design regimes.
When designing waterworks, the course of levels and discharges is calculated over a significant downstream length at various modes of operation, determined by the water content of the year, season, day of the week, and time of day.In this case, it is especially important to determine for various sections the values of both the maximum water levels (due to the danger of a breakthrough of structures) and the minimum ones (to ensure the normal functioning of water intakes, to prevent cavitation wear), as well as to determine the flow rates and flow rates in intermediate channels [1,2].In this paper, we consider advancing the filtration flow boundary depending on the time in a homogeneous soil massif limited by the slope of structures.

Materials and Methods
The article uses methods of physical modeling of structures made of reinforced soil.It develops the theoretical foundations of modeling with the introduction of the obtained data on full-scale structures.Graphical and analytical methods were used to check the main provisions of hydraulics based on the results of field and laboratory experiments on changing the boundary of the filtration flow over time [3,4].The parameters of the operation of the characteristics of water sources were analyzed in comparison with standard clean water and water with an elevated temperature regime and polluted with fin, dust, and mineral products.

Results and Discussion
Let the water level in front of the array change in time following the graph shown in Fig. 1.The soil mass is located on a horizontal aquiclude and is characterized by a filtration coefficient k.The movement of the filtration flow obeys the Darcy law [3].
The movement of the filtration flow boundary in time can be divided into two periods: 1) uniform rise, when , p t t d where p t is the time of water level rise; 2) the water level in front of the slope remains constant when Consider the first period of saturation of the soil massif with water.Let us assume that the boundary of the free surface of the filtration flow is a straight line moving in time parallel to itself [4,5].In this case, the movement of the filtration flow boundary can be determined by the movement of a separate boundary point.Let's take point C moving along the aquiclude (Fig. 2).where ‫ܥ‬ is dimensionless coefficient; ݇, ݊ are filtration coefficient and soil porosity; ‫ܪ‬ is pressure; ‫ݐ‬ is time.
With an instantaneous rise in the water level before the slope is rate of rise of the water level before the slope.
In a particular case, when the slope coefficient m=0, the following value of C was obtained: 1) for an instantaneous rise in the water level at C inst = 1.41 and C inst = 1.62 following studies [6,7]; 2) for a uniform rise in the water level С un = 1.Let's determine the value of C for the first period of saturation of the array with water at m≠0.
After squaring the left and right sides of equation ( 2) and transforming, we obtain .
Solving equations ( 3), (4), and ( 5), we obtain , 2 With a uniform rise in the water level, v x = const, and with an instantaneous rise and for the second period of saturation of the massif with water, vx is a variable value and depends on t [8,9].
Substituting expression (7) into equation ( 8), we obtain Separating the variables and integrating them respectively from 0 to x and from 0 to t, we obtain an equation for the case of an instantaneous rise in the water level in front of the slope Ht n k C x un 41 . 1 (9) here С i n s t = 1,41С un , and 1.41 is the C∞ coefficient.Previously, in the works of domestic and foreign authors, it was indicated that the value of С∞ can be equal in the range of 1.41 -1.62 [1,4].Assuming C∞=1.62, we obtain a solution with a certain margin for the instantaneous rise in the water level and the second period of saturation of the massif.
Following the research, with an instantaneous rise in the water level for m≠0, the change in the filtration flow boundary in time is expressed as follows: (10) where is the value of the dimensionless coefficient C at a distance x from the origin; Dividing the value of inst C in formula (10) by 1.41 or 1.62, we get The solution for the first filtering period will look like this: . a n k t C x un un un (12) The value un C in formula (12) can be obtained from the graph , constructed following dependence (3), or selected according to the same dependence (3), having previously substituted its value from formula (11)  Let us consider the calculation scheme (Fig. 3), where straight lines replace the curved streamlines; it is assumed that all of them are currently inclined at the same angle γ to the xaxis.This scheme quite accurately reflects reality.
On the other hand .
Equating the right parts of expressions ( 13) and ( 14), we obtain .To calculate the flow rate using formula (15), it is necessary to know the value of the angle γ.Let us determine γ using the maximum flow principle [11,12].To do this, we differentiate Q concerning γ and equate this expression to zero 0 cos sin cos cos cos sin sin sin sin sin cos Substituting expression ( 16) into formula (15), we obtain It can be seen from expression (16) that at E D angle γ=0, the streamlines will be horizontal.In reality, the streamlines will be horizontal at infinity when 0 E , i.e., the proposed method approximately determines the flow rate Q.However, for values E close to D S , Q is determined quite accurately, while Q will be calculated the more accurately

D S
, the closer to in value it is E .Q values were calculated using formula (17) for angles Substituting expressions (11) and (18) into formula (3), we finally obtain the following equations to determine the change in the filtration flow boundary in time:  19) and ( 20) are solved by selecting the values of x un or β.
Let us consider the second period of saturation of the massif with water [15,16].In this case, t ≥ t un and filtration occurs under a constant pressure H.
The speed v x will be determined by the dependence [17,18].
x  As seen from fig. 4, the discrepancy between the theoretical and experimental curves for changing the filtration flow boundary over time is no more than 4%.

Conclusions
1. Based on the above results of experiments on the saturation of the massif with water in a slotted tray, it can be concluded that they correspond to theoretical studies of changes in the boundary of the filtration flow in time [19,20].
2. The proposed method of calculation makes it possible to determine with a sufficient degree of accuracy the position of the boundary of the free surface of the filtration flow in time for the case of a change in the water level in front of the massif following the graph of the change in the water level in front of the slope.The equations obtained in this case are solved using the obtained data in tabular form.
3. The parameters of the first period of saturation of the massif with water (uniform rise) are calculated according to the equation for changing the boundary of the filtration flow in time, and the second period -according to the equation in an abbreviated form.

Fig. 1 .
Fig. 1.Graph of the change in water level before the slope.

Fig. 2 .
Fig. 2.Calculation scheme for the movement of groundwater.

3 )X
Consumption for saturation of the array with a uniform rise in the water level 2 is speed of propagation of the filtration flow boundary along the axis х.From expression (2), it follows that .

2 ,
instead of un C .However, in formula(11), the value un C can be calculated only for angles D we proceed as follows.

Fig. 3 .
Fig. 3. Calculation scheme for the movement of groundwater when replacing curved streamlines with straight lines According to Darcy's law, we have l H k X [9, 10].
Conferences 365, 03009 (2023) https://doi.org/10.1051/e3sconf/202336503009CONMECHYDRO -2022 data obtained for Q, following expression(6), the values of the dimensionless coefficient un C and the hydraulic gradient I were determined for various values of m[13,14].Intermediate values un C from С´р .о to С р.о can be obtained from a dependence similar to expression

(Fig. 4 .
Fig. 4. Graph of the dependence of the change in the boundary of the filtration flow in time ) (t f x for experiments: 1-№ 5; 2-№7.

Table 1 .
The values of the dimensionless coefficients С р.о , С р.о, and the corresponding hydraulic gradients I 0 , I 0 for different values of t

.
Separating the variables and integrating them from x un to x and from t un to t, we get I c taken from the table.To solve equation (23), it is necessary to preliminarily calculate inst С using formula